Comparison of Peak and RMS Gains for Discrete-Time Systems

A convolution system can have a frequency response which is small for all
frequencies, yet still greatly amplify the peaks of signals passing through it.
For finite-dimensional systems, the authors establish that the peak gain of a
system is not more than factor (2n+1) times more than the RMS gain of a system
(i.e., the maximum magnitude of its requency response), where n is the
dimension of the system. The bound implies that H_infty-optimal controllers,
which minimize the maximum of some disturbance-to-error transfer function,
cannot have very large peak gains from the disturbance to error.